Confinement of vorticity for the 2D Euler-α equations

Abstract

In this article we consider weak solutions of the Euler-α equations in the full plane. We take, as initial unfiltered vorticity, an arbitrary nonnegative, compactly supported, bounded Radon measure. Global well-posedness for the corresponding initial value problem is due M. Oliver and S. Shkoller. We show that, for all time, the support of the unfiltered vorticity is contained in a disk whose radius grows no faster than O((tlog⁡t)1/4). This result is an adaptation of the corresponding result for the incompressible 2D Euler equations with initial vorticity compactly supported, nonnegative, and p-th power integrable, p>2, due to D. Iftimie, T. Sideris and P. Gamblin and, independently, to Ph. Serfati.